Why is yaw damper inop limited to a maximum altitude?

[Luc Lion]

italia458,

you asked

Anyone have equations that explain this?

This equation is the equation of damped coupled oscillators.
It is somewhat complex as it is a tensorial equation:

M ḧ(t) + B ḣ(t) + K h(t) = 0

where M, B and K are tensors (read it as square matrices with as many columns or rows as there are oscillators) and ḧ,ḣ and h are matrices of 1 column by the oscillator count number of rows and each element of these h matrices are some function of the time t.

ḧ(t) is the acceleration vector. each row i contains the acceleration of the movement of oscillator i at time (t).
ḣ(t) is the speed vector. each row i contains the speed of the ocillator i at time (t).
h(t) is the displacement vector. each row i contains the displacement of the ocillator i at time (t).
The tensor M is the inertial tensor. Multiplied by ḧ(t), it gives the inertial pseudo-force. That is, each value in column i row j gives the inertial response in the degree freedom of oscillator j caused by the acceleration of oscillator i.
The tensor D is the damping tensor. It express the "friction" that damps the oscillation energy. The damping force vector is obtained by multiplication of D with the speed vector.
The tensor K is the stiffness tensor. It express the "reaction" force of the oscillator when multiplied with the displacement vector.

Dutch Roll is indeed a phenomenon of coupled oscillators.
One oscillator is the plane itself oscillating around its CG and pushed back in the relative wind direction by dynamic pressure on the rudder and rudder fin.
The second oscillator is the slip and roll oscillation fed by the slipstream pressure on swept wings or wings with dihedral.
There is much to say to explain why a single oscillator cannot generate a self sustained oscillation (even with energy fed by the dynamic pressure). The self sustained ocillation requires at least two coupled oscillators with different frequencies. But this is too long a story to fit here. If you are interested, please lurk into the links I provide below.

Back to coupled oscillators equation : when one or more of the oscillators are aerodynamic oscillators (at least the stiffness comes from an aerodynamic pressure), the equation becomes:

(Ms + ρπb² Ma) ḧ(t) + (Bs + ρvcπ Qb) ḣ(t) + (Ks + ρv²cπ Qk) h(t) = 0


The tensors Ms, Bs and Ks are just the structural part of the tensor, that is, the part that would exist if the oscillators were mere springs.
The terms ρπb²Ma , ρvcπQb and and ρv²cπQk are the aerodynamic part of these tensors.


ρπb²Ma is the aerodynamic inertia ; it is the mass of an arbitrary cylindar of air (with radius b) that moves with the oscillator movement. ρ is the density of the air.
ρvcπQb is the aerodynamic damping. v is the relative speed of the air flow and c is the width of the airfoil that oscillates (comes from c the chord in case of wing oscillation). Q is a very complex tensor and I shall not detail its contents.
ρv²cπQk is the aerodynamic force developped by the aerodynamic pressure. The factors ρv² is twice the dynamic pressure. cπ comes again from using the vibrating wing as the primary use case and Q is again a very compley tensor. It should be noted that this aerodynamic force depends on some incidence factors that are contained on the displacement vector h(t). The translation from oscillator displacement into a real lift factor is contained in the tensor Q.


Now, lets consider what happens when the altitude is increasing whilst keeping a constant indicated speed, and thus a constant dynamic pressure ρv²/2.
1. Obviously, the stiffness terms stays constant : it varies with ρv².
2. The aerodynamic inertia is going to decrease (ρ is decreasing) and thus the oscillation frequency will increase. However, Ma is usually small compared to Ms which stays constant. The frequency change is thus small.
3. However, the aerodynamic damping ρvcπQb is usually sizeable compared to the structural damping Bs. For dutch roll, the structural damping is insignificant compared to the aerodynamic one. The aerodynamic is proportional to ρv and not ρv². This means that, if ρ decreases and v increases so that ρv² stays constant, then ρv will decrease as the square root of the density decrease.


All in all, aerodynamic damping is decreasing with altitude.
That's why a mechanical damping may be absolutely necessary above a given altitude.

Note: the various flutter oscillations (aileron flutter, wing flutter, etc) are governed by the same equations and their prevention may require a speed or speed per altitude limitation.

Luc

Some links on coupled oscillators and aeroelasticity:
Ltas-aea ::Aeroelasticity Course
http://www.keybridgeti.com/videotrai..._Stability.PDF
http://www.cs.wright.edu/~jslater/SD...ter_banner.pdf
http://aerade.cranfield.ac.uk/ara/arc/rm/3011.pdf

Edit: when reviewing the post, I notice that I have not been very clear as to why the aerodynamic damping has a factor ρv and not ρv² or ρ.
Here it is :
- The aerodynamic pressure can have an effect that is directly opposed to and influenced by the oscillatory movement ; this is the lift on the aerofoil and also the aerodynamic stiffness. It is proportional to the dynamic pressure and is factored by ρv².
- The aerodynamic pressure also can have an effect that is perpendicular to the direction of the oscillatory movement. This is the drag generated by the aerofoil. Perpendicular means that, at first level of approximation, it makes no work on the oscillatory system. However, at higher levels of approximation, it does. And that is the damping.
Consider that ḣ(t) (the speed vector of the oscillatory movement) and v (the speed vector of the relative wind) are perpendicular. They sum up to give the wind speed relative to the oscillating surface. This composite speed is at an angle with the relative wind direction and the tangent of it is ḣ(t)/v.
(the total speed is square root of v²+ḣ(t)² and the sine of the angle is ḣ(t)/sqrt(v²+ḣ(t)²) )
Therefore, the drag force (which is always in the same direction as the relative wind) is also at an angle relative to the oscillating surface and its inclinaison is such that it opposes to the oscillatory movement (therefore the "damping").
The drag force is Fd = Cd.ρv²/2 (in the absence of the oscillatory movement)
and becomes Fd = Cd.ρ.(v²+ḣ(t)²)/2 when adding the oscillation.
The part of it that opposes to the oscillatory movement is given by the sine of the angle.
Damping = ( ḣ(t)/sqrt(v²+ḣ(t)²) ). Cd.ρ.(v²+ḣ(t)²)/2
Damping = Cd.ρ.( ḣ(t).sqrt(v²+ḣ(t)²) )/2
ḣ(t) is normally small compared to v ; therefore sqrt(v²+ḣ(t)²)can be safely simplified into v. We get:
Damping = ρv.ḣ(t).Cd/2
We see that this matches the aerodynamic damping term of the coupled oscillators equation (ρvcπ Qb) ḣ(t) where Cd/2 is replaced by cπ Qb.